Let Xt（x） be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt （x） be the Euler discretization scheme of SDEs with step 2-n . In this note, we prove that f...Let Xt（x） be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt （x） be the Euler discretization scheme of SDEs with step 2-n . In this note, we prove that for any R〉0 and γ∈（0, 1/2）, sup t∈[0,1],｜x｜≤R｜X nt （x,ω）-Xt （x,ω）｜≤ξR,γ（ω）2-nγ, n≥1, q.e., whereξR,γ（ω） is quasi-everywhere finite.展开更多
文摘Let Xt（x） be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt （x） be the Euler discretization scheme of SDEs with step 2-n . In this note, we prove that for any R〉0 and γ∈（0, 1/2）, sup t∈[0,1],｜x｜≤R｜X nt （x,ω）-Xt （x,ω）｜≤ξR,γ（ω）2-nγ, n≥1, q.e., whereξR,γ（ω） is quasi-everywhere finite.
